NONLINEAR BACKSTEPPING CONTROL WITH OBSERVER DESIGN FOR A 4 ROTORS HELICOPTER L. Mederreg, F. Diaz and N. K. M’sirdi LRV, Laboratory of Robotics of Versailles, Université de Versailles Saint Quentin en Yvelines, 10, avenue de l’Europe 78140, Vélizy, France. [email protected], [email protected] Abstract: This paper develops a nonlinear observer and controller applied to a helicopter with 4 rotors. We show how to design a powerful nonlinear control law in term of robustness. In addition, this technique allows reducing the number of sensors to be embedded in the flying system by observing non available or non measurable entities. Performances and the stability of the suggested controller are analyzed through simulations carried out on the model (kinematics and dynamic equations). Keywords: System modelling, non linear controller, Backstepping Observes and controllers.

1. INTRODUCTION The miniature and autonomous air machines arouse a growing interest in the civil and military domain. The fields of application of these apparatuses are vast. One can state the ecological exploration mission, air cartography, and arts structure auscultation. The miniature helicopters are particularly suitable for this kind of applications. They are easy to handle, able to make hovering and can take off or land vertically. Moreover, their configuration appears particularly interesting for some applications, where stability, reliability, and the safety of the people and the goods are of primary importance. Indeed, this helicopter has no “head of rotor” and is mechanically very simple. It has also a weak catch with the wind; this increases its stability in disturbed environment. However, due to the complexity of piloting the helicopters under unfavourable climatic conditions obliges to avoid an open loop control. Consequently, to achieve high level missions planned by human operators, it is necessary to design control laws able to track predefined trajectories even in the presence of wind and turbulences. This idea motivates many studies towards the helicopters [1][2][7][5][9]. An exact linearization method has been developed on the kino-dynamic model with four rotors [11] [12]. However, this technique encounters problems of robustness in presence of disturbances and errors of the system parameters estimations. Moreover, it requires measurements of the full system state vector. In order to overcome these disadvantages, we directed our efforts towards an approach of nonlinear control that guarantees the properties of robustness and reduces also the number of required sensors, by observing states of the system which are difficult to measure. The method proposed

is based on the Backstepping approch. It consists of the use of a triangular structure of the given model for which the feed back is computed step by step where a gain is added to compensate the errors. 2. THE DYNAMICAL MODEL OF THE HELICOPTER In this section, we develop the kinematics dynamical model of a 4 rotors helicopter. The 4 rotors are horizontal and the configuration is symmetric. The rotors’ speeds can be controlled independently. Two rotors turn in the clock wise direction whereas the two others turn in the opposite direction. This allows avoiding the couple effects on the platform. A forward motion can be obtained by accelerating the speed of the rear rotor and reducing the speed of the front one. The lateral motion can be obtained by the same manner using left and right rotors. The yaw control is obtained by accelerating two “front to front” rotors and slowing the two others. The four rotors helicopter is assumed to be rigid body, having six degrees of freedom, and subject to external efforts. The mathematical model contains kinematics and dynamic equations. The kinematics equations describe the relation between the position and the orientation of the helicopter and its speed. The absolute position of the helicopter is given by the three co-ordinates ( x0 , y0 , z0 ) of its centre of gravity, with respect to an inertial reference frame attached to the ground and its orientation by the three Euler angles (ψ , θ , φ ) . These angles are called, the yaw ( − π 0 u → α ⇒ V ≤ − k e 2 ≤ 0 ⇒ e → 0 0

11

1

11 11

11

To force u0 to tend to α11 , we define a new error to be cancelled: e12 = u0 − α11 We choose a new LYAPUNOV candidate function: 1 V2 = e122 ⇒ V2 = e12 e12 = e12 (u0 − α11 ) 2 We notice that no entry of the system appears yet. We carry on by introducing a virtual input to cancel the tracking error. The input of the system will appear in the fourth iteration. We obtain a differential equation with four unknowns. An equivalent procedure is to be applied to the three remaining outputs. We will obtain four differential equations with four unknowns which are: (u1 , u2 , u3 , u4 ) . By solving the system, we obtain the expressions of the controller. 3.3 Synthesis of a Bakstepping control law with observer In this part, we consider that the absolute velocities are not measurable. We shall develop a controller with an observer by considering that parameters of the system are known. The objective is to control the absolute position and the yaw angle of the helicopter by observing the absolute velocity ( uˆ O ;vˆ O ; wˆ O ) which is difficult to measure with good precision. We consider that the remaining state variables are measured. From the system (6), we will launch a procedure allowing to

control the four outputs ( x0 , y0 , z0 ,ψ ) and to cancel

B1 ( x ) u1 + B2 ( x ) u2 + B3 ( x ) u3 + B4 ( x ) u4 =

0 ) . the observation errors (u0 , v0 , w

− c8 z8 − z7 − B5 ( x , xˆ ) − d8 B6 ( x ) z8 2

Step 1: We want to track the trajectory xd . We define the error z1 = x0 − xd : and we want to make it tend to zero. By deriving the error, one will obtain: z1 = uˆ0 + u0 − xd

The expression of the controller did not appear. We introduce thus a virtual input such as: α1 = − c1 z1 + xd − d1 z1

Let us note that the tuning gains di >0 allow us to adjust the observation errors and the gains ci >0 affect the system stability. At the next step, we define the error z2 = uˆ0 − α1 to be cancelled. By applying the same procedure, we notice that the inputs don’t appear yet. They will appear at the fourth iteration. Step 4: The error to be cancelled is: z4 = (− γ 1 K1 / m + γ 2 γ 1 + 1) uˆ0 − α 3 . By driving it will obtain: z 4 = A1 ( x ) u1 + A2 ( x ) u 2 + A3 ( x ) u3 + A4 ( x ) u 4

+ A5 ( x , xˆ ) + ϕ 4 ( x ) u0 At this step, we can see the inputs of the system in the expression. Thus we stop the procedure since we have the expressions of the introduced errors z1 , z2 , z3 , z4 in function in function of the inputs.

The obtained equation is: A1 ( x ) u1 + A2 ( x ) u2 + A3 ( x ) u3 + A4 ( x ) u4 = 2 − c4 z4 − z3 − A5 ( x , xˆ ) − d 4 ϕ 4 ( x ) z4

By using this relation, we will obtain the following system (with A0=-K1/m): z1 z2 z3 z4

= = = =

− − − −

c1 z1 c2 z2 c3 z3 c4 z4

+ − − −

2 d1 ϕ1 z1

+ ϕ1 u0 − 2 + z3 + ϕ 2 u0 − d 2 ϕ 2 z2 2 + z4 + φ3 u0 − d3 ϕ 3 z3 2 + ϕ 4 ( x ) u0 − d 4 ϕ 4 ( x ) z4

z2 z1 z2 z3

Moreover, knowing that the dynamic error of the observer is given by: u0 = A0 u0 , ( A0 < 0 ), and by defining a LYAPUNOV candidate function 4

V1 =

∑( 2z 1

2 i

1

+

di

i =1

2 P0 u0 ) ≥ 0 .

V1 ≤ −

∑( i =1

ci zi + 2

2 − c12 z12 − z11 − C5 ( x , xˆ ) − d12 C6 ( x ) z12

0 u1 + 0 u2 + D3 ( x ) u3 + D4 ( x ) u4 = − c14 z14 − z13 − D5 ( x )

Knowing the errors dynamic of observing v0 , w 0 , and by defining three other LYAPUNOV candidates (P1=m/2K2, P2=m/2K3) 8

∑( 2z

V2 =

1

i=5

12

V3 =

∑ i=9

(

1 2

zi +

1

2

di

2 i

+

1 di

2 P1 v0 )

2 P2 w 0 ) & V4 =

1 2

14

∑z

2 i

.

i = 13

We can show that the errors zi , v0 , w 0 converge to zero. Hence, the equilibrium: ( zi = 0, u0 = 0, v0 = 0, w0 = 0 ) is globally stable. From the previous equations, we end up with a system of 4 equations of the form A × U = H where the unknowns are the inputs of the system u1, u2, u3, u4: ⎡ u1 ⎢ ⎢ u2 ⎢ ⎢ u3 ⎢ ⎣ u4

⎡ A1 ( x ) ⎤ ⎢ ⎥ ⎢ B1 ( x ) ⎥ ⎥ = ⎢ ⎢ C1 ( x ) ⎥ ⎢ ⎥ ⎢⎣ 0 ⎦ ⎡ − c4 z 4 ⎢ ⎢ − c8 z8 ⎢ ⎢ − c12 z12 ⎢ ⎢⎣

A2 ( x )

A3 ( x )

B2 ( x )

B3 ( x )

C2 ( x )

C3 ( x )

0

D3 ( x )

A4 ( x ) ⎤

−1

⎥ B4 ( x ) ⎥ ⎥ C4 ( x ) ⎥ ⎥ D4 ( x ) ⎦⎥

×

2 − z 3 − A5 ( x , xˆ ) − d 4 ϕ 4 ( x ) z 4

− z7

2 − B5 ( x , xˆ ) − d 8 ϕ 8 ( x ) z8

− z11 − C 5 ( x , xˆ ) − d12 ϕ12 ( x ) z12 2

− c14 z14 − z13 − D5 ( x )

(7) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

The system has a finite solution unless in certain cases where the matrix A is singular which describes unreachable positions or configuration of the helicopter. 4.SIMULATIONS AND RESULTS In order to evaluate the performance of the proposed controller, we performed some simulations. The tracked trajectory is a vertical helix given by the following expressions: x d = cos(t ) ; y d = sin( t / 2) ; z d = − t / 10 ; ψ d = π / 3 The initial positions of the helicopter are set to:

We obtain (with P0=-1/2A0): 4

C1 ( x ) u1 + C2 ( x ) u2 + C3 ( x ) u3 + C4 ( x ) u4 =

3 u0

x0 = 0 , y0 = 0 , z 0 = 1, ψ = 0

2 ∞

) ≤ 0

4 di

We conclude that the errors zi , u0 converge to zero by proving the global stability of the equilibrium ( zi = 0 , u0 = 0 ) . The same procedure is to be applied to the 3 remaining tracking. We shall obtain three additional equations:

The known parameters are: m = 0.6 , K i = 3, g = 9.81 We simulate also the perturbation due to the wind blowing; the strength is set to 7 Newton. The gain of the controller is set to: ci = 50 , di = 1.5 . The figure 2 represents the tracking errors of the controller considering that the state vector is completely measurable. We notice that the tracking is effective, and it takes about 0.4 sec to the four outputs to converge. The

controller shows a good robustness to the simulated perturbation. The figure 3 illustrates the observation errors

u0 , v0 , w 0 of the absolute velocity of the helicopter. We notice that the observation is good since the errors tend to zero after a short time of response (about 0.25 sec).

5. CONCLUSION In this paper a dynamical model of a four rotors helicopter has been presented. For such model, we showed that it is possible to develop a non linear controller with observers based on the Backstepping. This approach has shown a good robustness of the controller. It has also permit to reduce the number of the required sensors that have to be used to elaborate the control law, by observing the absolute velocity of the helicopter which is difficult to measure. Intensive simulations have been done to validate the performance and the stability of the controller. An adaptive part is under development. It will allow the estimation of the platform parameters, which are not well-known and may change when operating. REFERENCES

Fig. 2: Tracking errors

0 Fig. 3: Observation errors u0 , v0 , w

Fig. 4: Tracking errors for Observer based control The figure 4 shows that the system converges to the desired trajectory even though it takes more time in compare to the first controller. This is due to time response of the observer.

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